CHAPTER • Production 219 units to 3, and then declines to 2/3 and to 1/3 Clearly, as more and more labor replaces capital, labor becomes less productive and capital becomes relatively more productive Therefore, we need less capital to keep output constant, and the isoquant becomes flatter DIMINISHING MRTS We assume that there is a diminishing MRTS In other words, the MRTS falls as we move down along an isoquant The mathematical implication is that isoquants, like indifference curves, are convex, or bowed inward This is indeed the case for most production technologies The diminishing MRTS tells us that the productivity of any one input is limited As more and more labor is added to the production process in place of capital, the productivity of labor falls Similarly, when more capital is added in place of labor, the productivity of capital falls Production needs a balanced mix of both inputs As our discussion has just suggested, the MRTS is closely related to the marginal products of labor MPL and capital MPK To see how, imagine adding some labor and reducing the amount of capital sufficient to keep output constant The additional output resulting from the increased labor input is equal to the additional output per unit of additional labor (the marginal product of labor) times the number of units of additional labor: In §3.1, we explain that an indifference curve is convex if the marginal rate of substitution diminishes as we move down along the curve Additional output from increased use of labor = (MPL)(⌬L) Similarly, the decrease in output resulting from the reduction in capital is the loss of output per unit reduction in capital (the marginal product of capital) times the number of units of capital reduction: Reduction in output from decreased use of capital = (MPK)(⌬K) Because we are keeping output constant by moving along an isoquant, the total change in output must be zero Thus, (MPL)(⌬L) + (MPK)(⌬K) = Now, by rearranging terms we see that (MPL)/(MPK) = -(⌬K/⌬L) = MRTS (6.2) Equation (6.2) tells us that the marginal rate of technical substitution between two inputs is equal to the ratio of the marginal products of the inputs This formula will be useful when we look at the firm’s cost-minimizing choice of inputs in Chapter Production Functions—Two Special Cases Two extreme cases of production functions show the possible range of input substitution in the production process In the first case, shown in Figure 6.7, inputs to production are perfect substitutes for one another Here the MRTS is constant at all points on an isoquant As a result, the same output (say q3) can be produced with mostly capital (at A), with mostly labor (at C), or with a balanced combination of both (at B) For example, musical instruments can be manufactured almost entirely with machine tools or with very few tools and highly skilled labor Figure 6.8 illustrates the opposite extreme, the fixed-proportions production function, sometimes called a Leontief production function In this case, In §3.1, we explain that two goods are perfect substitutes if the marginal rate of substitution of one for the other is a constant • fixed-proportions production function Production function with L-shaped isoquants, so that only one combination of labor and capital can be used to produce each level of output